Mathematics Question and Solution
Let the side length of the regular octagon be 1 unit.
a = ⅛(180(8-2))
a = ⅛(180*6)
a = 3*45
a = 135°
a is the single interior angle of the regular octagon.
b = ½(360-135-135)
b = ½(360-270)
b = ½(90)
b = 45°
c = ½(180-135)
c = ½(45)
c = 22.5°
d = 135-45-22.5
d = 90-22.5
d = 67.5°
e = (180(5-2))
e = (180*3)
e = 540°
e is the sum of interior angles of a regular pentagon.
f = ½(540-3(135))
f = ½(540-405)
f = ½(135)
f = 67.5°
g = f-c
g = 67.5-22.5
g = 45°
h = 180-d-g
h = 180-67.5-45
h = 135-67.5
h = 67.5°
i = 135-f
i = 135-67.5
i = 67.5°
Observing Cosine Rule.
j² = 1²+1²-2*1*1cos135
j² = 2-2cos135
j = 1.847759065 units.
k² = 1.847759065²+12-2*1*1.847759065cos67.5
k = 1.7320508075 units.
It implies, the required angle x is;
Observing Sine Rule.
(1.7320508075/sin67.5)/(1/sinx)
x = 32.2356103176°
x ≈ 32.24° to 2 decimal places degree.
